2022/10/24 by Glück, Jochen, Mironchenko, Andrii
#37L15 #47A10 #47B65 #47D06 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2210.13566
We consider generators of positive C0-semigroups and, more generally, resolvent positive operators A on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound s(A). Our main result characterizes s(A) < 0 in terms of so-called small-gain conditions that describe the behaviour of Ax for positive vectors x. This is new even in case that the underlying space is an Lp-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property s(A) < 0 if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of A is negative. To treat the latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent positive operators. When A is the generator of a positive C0-semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case. In the same vein, we prove a Collatz--Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups.